Contents

Idea

A classifying space for some sort of data refers to a space (or a more general object), usually written ℬ(data)\mathcal{B}(data), such that maps X→ℬ(data)X\to \mathcal{B}(data) correspond to data over XX.

The classical example is the classifying space ℬG\mathcal{B}G of a groupGG, which has the property that there is a bijection between homotopy classes of maps X→ℬGX\to \mathcal{B}G and isomorphism classes of GG-bundles over XX. (In fact, one can jack this up to an equivalence of groupoids or ∞\infty-groupoids.) Various improvements of this are possible which classify bundles with extra structure or fibrations.

Categorically, the corresponding statement is that Grothendieck fibrations over a categoryXX correspond to pseudofunctorsXop→CatX^{op}\to Cat. Thus CatCat is the “classifying space for categories.” Similarly, discrete fibrations over XX correspond to functorsXop→SetX^{op}\to Set.

To see the connection between the two, consider the case when XX is a groupoid and we restrict the fibers of the fibration to be isomorphic to a given set FF. Then the functor Xop→SetX^{op}\to Set must land in the subcategory of SetSet consisting of just the automorphisms of FF, which is the one-object groupoid corresponding to the automorphism groupAut(F)Aut(F). If we further restrict the automorphisms appearing to preserve some given structure on FF, so that they lie in some smaller group GG, then the “classifying space” will be the one-object groupoid corresponding to GG. Under the homotopy hypothesis, groupoids correspond to homotopy 1-types, and the one-object groupoid of a group GG corresponds precisely to the usual topological classifying space ℬG\mathcal{B}G (in fact, this is one construction of ℬG\mathcal{B}G). For this reason, BG\mathbf{B}G is often used to denote that one-object groupoid; see the delooping hypothesis and the discussion at category algebra.

The phrase “classifying space” is also sometimes used for the realization of the nerve of any category, although it is more complicated to say what exactly this space “classifies.” (One answer is “torsors modulo concordance.”)

For orthogonal and unitary structure groups

For G=O(n)G = O(n) the orthogonal group and G=U(n)G = U(n) the unitary group, there are standard realizations of the corresponding classifying spaces as direct limits of Grassmannian spaces. (See for instance (May, p. 196), where some of the following is taken from).

Let Vn(ℝq)V_n(\mathbb{R}^q) be the Stiefel variety? of orthonormal nn-frames in the Cartesian spaceℝq\mathbb{R}^q. Its points are nn-tuples of orthonormal vectors in ℝq\mathbb{R}^q, and it is topologized as a subspace of (ℝq)n(\mathbb{R}^q)^n, or, equivalently, as a subspace of (Sq−1)n(S^{q-1})^n. It is a compactmanifold.

Let Gn(ℝq)G_n(\mathbb{R}^q) be the Grassmannian of nn-planes in ℝq\mathbb{R}^q. Its points are the n-dimensional subspaces of ℝq\mathbb{R}^q. Sending an nn-tuple of orthonormal vectors to the nn-plane they span gives a surjective function Vn(ℝq)→Gn(ℝq)V_n(\mathbb{R}^q) \to G_n(\mathbb{R}^q), and we topologize Gn(ℝq)G_n(\mathbb{R}^q) as a quotient space of Vn(ℝq)V_n(\mathbb{R}^q). It too is a compact manifold.

The standard inclusion of ℝq\mathbb{R}^q in ℝq+1\mathbb{R}^{q+1} induces inclusions Vn(ℝq)↪Vn(ℝq+1)V_n(\mathbb{R}^q) \hookrightarrow V_n(\mathbb{R}^{q+1}) and Gn(ℝq)↪Gn(ℝq+1)G_n(\mathbb{R}^q) \hookrightarrow G_n(\mathbb{R}^{q+1}). We define Vn(ℝ∞)V_n(\mathbb{R}^\infty) and Gn(ℝ∞)G_n(\mathbb{R}^\infty) to be the unions of the Vn(ℝq)V_n(\mathbb{R}^q) and Gn(ℝq)G_n(\mathbb{R}^q), with the topology of the union.

Then Gn(ℝ∞)G_n(\mathbb{R}^\infty) is a model for the classifying space BO(n)B O(n).

Composition with a forgetful functor U:(topologicaldata)→(topologicalspaces)U: (topological data) \to (topological spaces) gives a classifying space. In such cases one would also like a homotopically defined functor

there are notions of homotopy for both types of data leading to a bijection of homotopy classes for some XX

[X,UℬC]≅[ΞX*,C].[X,U\mathcal{B}C] \cong [\Xi X_*, C].

This happens for the algebraic data of crossed complexes and the topological data of filtered spaces, when XX is a CW-complex, and Ξ\Xi is the fundamental crossed complex of a filtered space. Thus in this case the classifying space does classify homotopy classes of maps, and more work is needed to sort out the data over XX which this classifies (gerbes?).

However ℬC\mathcal{B}C is in this case defined by a nerve construction which generalises that for groupoids, and can also be applied to topological crossed crossed complexes, giving a simplicial space.

Mike: I don’t really get any intuition from that. There might be lots of functors from “algebraic data” to “topological data” but it seems to me that only particular sorts of them deserve the name “classifying space.” Can you say more specifically what sorts of functors you have in mind, and relate it to the more basic ideas that I am familiar with? What do these classifying spaces classify?

Ronnie What I am trying to characterise is that higher categories carry structure such as a filtration by lower dimensional higher categories, or, for multiple structures, a multiple filtration. Thus one expects a classifying space to inherit this extra structure. Conversely, the construction of an infinity-groupoid from a space might depend on this extra structure.

So I spent 9 years trying to construct a strict homotopy double groupoid of a space, yet Philip Higgins and I did this overnight in 1974 when we tried the simplest relative example we could think of: take homotopy classes of maps from a square to XX which take the edges to a subspace X1X_1 and the vertices to a base point x0x_0. Then the filtered case took another 4 years or so to complete.

Then Loday constructed a cat-n-group from an n-cube of spaces, published in 1982. Its multi-nerve is an (n+1)(n+1)-simplicial set, whose realisation is (n+1)(n+1)-filtered.

A strict homotopy double groupoid of a Hausdorff space has been constructed but this needs a subtle notion of thin homotopy.

Of course the filtration for a group is not so apparent, but it is more clear that groupoids carry structure in dimension 0 and 1, and hence are useful for representing non connected homotopy 1-types, and their identifications in dimension 0, as explained in the first edition (1968) of my Topology book.

Some such constructions arise from generalisations of the Dold-Kan correspondence, with values in simplicial sets. For example, from a crossed complexCC one obtains a simplicial set Nerve(C)Nerve(C) which in dimension nn is Crs(Π(Δ*n),C)Crs(\Pi(\Delta^n_*),C). The geometric realisation ℬC\mathcal{B}C of this is canonically filtered by the skeleta of CC, so ℬ\mathcal{B} is really a functor to filtered spaces. This ties in with the functor Π\Pi which goes in the opposite direction. But note that there is a different filtration of the space ℬC\mathcal{B}C since it is a CW-complex, and so Π\Pi of this filtration gives a free crossed complex.

Special cases of crossed complexes are groupoids, and so we get the classifying space of a groupoid; and similarly of a crossed module.

A crossed module is equivalent to a category object in groups, and so a nerve of this can be constructed as a bisimplicial set. The geometric realisation of this is naturally bifiltered, in several ways!

In considering what is desirable for a fundamental infinity-groupoid one should bring the notion of classifying space, and its inherited structure, into account.

For simplicial groups

The W¯(−)\bar W(-)-construction (see simplicial group and groupoid object in an (∞,1)-category) which gives the classifying space functor for simplicial groups and simplicially enriched groupoids is given in the entry on simplicial groups. It provides a good example of the above as the W-bar functor is right adjoint to the Dwyer-Kan loop groupoid functor and induces an equivalence of homotopy categories between that of simplicial sets and that of simplicially enriched groupoids. The simplicial sets here are playing the role of ‘topological data’.